beth numbers

The beth numbers are infiniteMathworldPlanetmathPlanetmath cardinal numbersMathworldPlanetmath defined in a similar manner to the aleph numbers, as described below. They are written α, where is beth, the second letter of the Hebrew alphabet, and α is an ordinal numberMathworldPlanetmath.

We define 0 to be the first infinite cardinal (that is, 0). For each ordinal α, we define α+1=2α. For each limit ordinalMathworldPlanetmath δ, we define δ=αδα.

Note that 1 is the cardinality of the continuumMathworldPlanetmathPlanetmath.

For any ordinal α the inequality αα holds. The Generalized Continuum Hypothesis is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the assertion that α=α for every ordinal α.

For every limit ordinal δ, the cardinal δ is a strong limit cardinal. Every uncountable strong limit cardinal arises in this way.

Title beth numbers
Canonical name BethNumbers
Date of creation 2013-03-22 14:17:09
Last modified on 2013-03-22 14:17:09
Owner yark (2760)
Last modified by yark (2760)
Numerical id 9
Author yark (2760)
Entry type Definition
Classification msc 03E10
Related topic AlephNumbers
Related topic GeneralizedContinuumHypothesis